Answer :
Let's break down the step-by-step solution to the problem using the provided data from the table.
#### Given Information:
The table gives us the joint probabilities and total probabilities for blood types A, B, AB, and O with either positive (Pos.) or negative (Neg.) Rh factors.
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline & A & B & AB & O & \text{Total} \\ \hline \text{Neg.} & 0.07 & 0.02 & 0.01 & 0.08 & 0.18 \\ \hline \text{Pos.} & 0.33 & 0.09 & 0.03 & 0.37 & 0.82 \\ \hline \text{Total} & 0.40 & 0.11 & 0.04 & 0.45 & 1.0 \\ \hline \end{array} \][/tex]
#### To Find:
1. The probability that a person has a positive Rh factor given that he/she has type O blood, [tex]\( P(\text{Pos. } | \text{ O}) \)[/tex].
2. The true comparison for the statement "There is a greater probability for a person to have a [Type] than a person to have a positive Rh factor given type O blood."
#### Solution for [tex]\( P(\text{Pos. } | \text{ O}) \)[/tex]:
We need to find the conditional probability, which can be calculated using the formula:
[tex]\[ P(\text{Pos. } | \text{ O}) = \frac{P(\text{Pos. and O})}{P(\text{O})} \][/tex]
From the table:
- [tex]\( P(\text{Pos. and O}) = 0.37 \)[/tex]
- [tex]\( P(\text{O}) = 0.45 \)[/tex]
So:
[tex]\[ P(\text{Pos. } | \text{ O}) = \frac{0.37}{0.45} \approx 0.8222 \][/tex]
Thus, the probability that a person has a positive Rh factor given that he/she has type O blood is approximately [tex]\( 0.8222 \)[/tex] or [tex]\( 82.22\% \)[/tex].
#### Comparison Statement:
The calculated probability [tex]\( P(\text{Pos. } | \text{ O}) \)[/tex] is 82.22%. We need to check this against the total probabilities of other types to find a match with “greater probability”.
[tex]\[ \begin{array}{c|c} \text{Blood Type} & \text{Total Probability} \\ \hline \text{A} & 0.40 \\ \text{B} & 0.11 \\ \text{AB} & 0.04 \\ \text{O} & 0.45 \\ \end{array} \][/tex]
Comparing 82.22%:
- [tex]\( P(\text{A}) = 0.40 \)[/tex] which is 40%
- [tex]\( P(\text{B}) = 0.11 \)[/tex] which is 11%
- [tex]\( P(\text{AB}) = 0.04 \)[/tex] which is 4%
- [tex]\( P(\text{O}) = 0.45 \)[/tex] which is 45%
Clearly, [tex]\( 45\% > 82.22\% \)[/tex].
Hence, we can complete the statement:
The probability that a person has a positive Rh factor given that he/she has type [tex]$O$[/tex] blood is [tex]$82.22\%$[/tex]. There is a greater probability for a person to have type [tex]$O$[/tex] blood than a person to have a positive Rh factor given type [tex]$O$[/tex] blood.
#### Given Information:
The table gives us the joint probabilities and total probabilities for blood types A, B, AB, and O with either positive (Pos.) or negative (Neg.) Rh factors.
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline & A & B & AB & O & \text{Total} \\ \hline \text{Neg.} & 0.07 & 0.02 & 0.01 & 0.08 & 0.18 \\ \hline \text{Pos.} & 0.33 & 0.09 & 0.03 & 0.37 & 0.82 \\ \hline \text{Total} & 0.40 & 0.11 & 0.04 & 0.45 & 1.0 \\ \hline \end{array} \][/tex]
#### To Find:
1. The probability that a person has a positive Rh factor given that he/she has type O blood, [tex]\( P(\text{Pos. } | \text{ O}) \)[/tex].
2. The true comparison for the statement "There is a greater probability for a person to have a [Type] than a person to have a positive Rh factor given type O blood."
#### Solution for [tex]\( P(\text{Pos. } | \text{ O}) \)[/tex]:
We need to find the conditional probability, which can be calculated using the formula:
[tex]\[ P(\text{Pos. } | \text{ O}) = \frac{P(\text{Pos. and O})}{P(\text{O})} \][/tex]
From the table:
- [tex]\( P(\text{Pos. and O}) = 0.37 \)[/tex]
- [tex]\( P(\text{O}) = 0.45 \)[/tex]
So:
[tex]\[ P(\text{Pos. } | \text{ O}) = \frac{0.37}{0.45} \approx 0.8222 \][/tex]
Thus, the probability that a person has a positive Rh factor given that he/she has type O blood is approximately [tex]\( 0.8222 \)[/tex] or [tex]\( 82.22\% \)[/tex].
#### Comparison Statement:
The calculated probability [tex]\( P(\text{Pos. } | \text{ O}) \)[/tex] is 82.22%. We need to check this against the total probabilities of other types to find a match with “greater probability”.
[tex]\[ \begin{array}{c|c} \text{Blood Type} & \text{Total Probability} \\ \hline \text{A} & 0.40 \\ \text{B} & 0.11 \\ \text{AB} & 0.04 \\ \text{O} & 0.45 \\ \end{array} \][/tex]
Comparing 82.22%:
- [tex]\( P(\text{A}) = 0.40 \)[/tex] which is 40%
- [tex]\( P(\text{B}) = 0.11 \)[/tex] which is 11%
- [tex]\( P(\text{AB}) = 0.04 \)[/tex] which is 4%
- [tex]\( P(\text{O}) = 0.45 \)[/tex] which is 45%
Clearly, [tex]\( 45\% > 82.22\% \)[/tex].
Hence, we can complete the statement:
The probability that a person has a positive Rh factor given that he/she has type [tex]$O$[/tex] blood is [tex]$82.22\%$[/tex]. There is a greater probability for a person to have type [tex]$O$[/tex] blood than a person to have a positive Rh factor given type [tex]$O$[/tex] blood.
